"PET is working on a theory that the perpetrator could have been inspired by the events in Paris," Madsen said, according to the AP. "He could also have been inspired by material sent out by (ISIS) and others."

Police said the suspect opened fire on officers as they were monitoring an address in Norrebro while investigating the two shootings. Authorities did not identify the suspect, who was killed when police returned fire.

The firefight capped an extensive manhunt underway following the first reports of gunshots near an event hosting controversial Swedish cartoonist Lars Vilks. One person was killed and three police officers were wounded in the attack near the "Arts, Blasphemy and Freedom of Expression" event at 4 p.m. Saturday (10 a.m. E.T.) at Krudttonden cafe in the Osterbro district.

Vilks, 68, has been the subject of death threats for caricaturing the Prophet Muhammad in 2007. He was not injured in the shooting, which the Danish prime minister described as a "terrorist attack."

Hours later, in the early hours of Sunday morning, a gunman shot a civilian and two police officers outside of a synagogue. The civilian later died.

"We are working under the assumption that it was the same perpetrator who was behind both shooting incidents," said Police Commissioner Torben Molgaard Jensen. "And we are also working under the assumption that the perpetrator who was shot by SWAT police...is the person who carried out these attacks."

Vilks told The Associated Press that he believes he was the target of the shooting.

"What other motive could there be?" he said, and referred to the attacks on the offices of the French satirical magazine Charlie Hebdo in Paris in January.

Vilks has been threatened before: a Pennsylvania woman last year got a 10-year prison term for a plot to kill the cartoonist. In 2010, two brothers tried to burn down his house in southern Sweden and were imprisoned for attempted arson.

Danish Prime Minister Helle Thorning-Schmidt put the entire nation on high alert and on Sunday visited the scene of the synagogue shooting.

"Denmark has been hit by terror," she said. "We do not know the motive for the alleged perpetrator's actions, but we know that there are forces that want to hurt Denmark. They want to rebuke our freedom of speech."

The U.S. condemned the shootings and said it was ready to help the Danish government investigate, if asked.

"We offer our condolences to the loved ones of the deceased victim, and our thoughts are with those wounded in this attack," U.S. National Security Council spokesperson Bernadette Meehan said.